11.The 11th term and the 21st term of an A.P. are 16 and 29 respectively. Find1.the 1st term and the common differenceii.the 34th termiii.'n' such that tn = 55.​

i. To find the first term (a) and the common difference (d) of an arithmetic progression (A.P.), we can use the formula for the nth term of an A.P., which is a + (n-1)d.

Given that the 11th term is 16 and the 21st term is 29, we can set up the following two equations:

16 = a + 10d (equation 1) 29 = a + 20d (equation 2)

Subtracting equation 1 from equation 2 gives us:

13 = 10d

So, d = 1.3

Substituting d = 1.3 into equation 1 gives us:

16 = a + 10(1.3) 16 = a + 13 a = 16 - 13 a = 3

ii. To find the 34th term, we can use the formula for the nth term of an A.P.:

Tn = a + (n-1)d T34 = 3 + (34-1)(1.3) T34 = 3 + 33(1.3) T34 = 3 + 42.9 T34 = 45.9

iii. To find 'n' such that Tn = 55, we can set up the following equation and solve for n:

55 = 3 + (n-1)(1.3) 52 = (n-1)(1.3) n-1 = 52/1.3 n-1 = 40 n = 41

So, the 1st term and the common difference are 3 and 1.3 respectively, the 34th term is 45.9, and 'n' such that Tn = 55 is 41.